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Some centrality measures can be interpreted in terms of walks.

  1. Degree centrality relates to a walk of length one: The more walks of length one reach a node, the higher this centrality measure.
  2. Betweenness centrality relates to walks along the shortest paths. The higher the number of shortest paths that pass through a node, the higher this centrality measure.
  3. Eigenvector centrality relates to a walk of infinite length according to the Wikipedia entry about centrality measures.

I find this characterization of these centrality measures quite useful but I have difficulties developing an intuition why eigenvector centrality relates to walk of infinite length. Any idea?

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Eigenvector centrality relates to a random walk of infinite length, in which each next node is chosen uniformly at random from the set of neighbours of the current node (in an unweighted network; in a weighted network, the next node is chosen with a probability proportional to the edge weight). The eigenvector centrality of a node is proportional to the frequency with which that node is visited during such a walk (corresponding to the stationary distribution of a Markov chain). IMO, the key attribute of this walk is "random", not "infinite" (a random walk with a long but finite length usually comes pretty close to approximate the eigenvector centrality).

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